On the decomposition of the tensor algebra of the classical Lie algebras
type: Item
instance of: scholarly article
title: On the decomposition of the tensor algebra of the classical Lie algebras
zbMATH Open document ID: 0577.17003
DOI: 10.1016/0001-8708(85)90035-0
published in: Advances in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(T_ n\) denote the tensor algebra of the Lie algebra \(gl_ n({\mathbb{C}})\), \(T^ f_ n\) its f-graded piece, \(R_{\infty}\) an appropriately defined sequence of irreducible representations \(R_ n\) of \(gl_ n({\mathbb{C}})\), \(m(R_ n,\psi,f)\) the multiplicity of \(R_ n\) in \(T^ f_{n \psi}\), the \(\psi\)-isotypic component for an irreducible character. Recently \textit{R. K. Gupta} and \textit{R. P. Stanley} got striking results on the limit behavior of the multiplicities as \(n\to \infty\) (their papers are not published yet) for the case of the symmetric algebra. The author shows that the limit exists and has a very simple expression. His results include the cases of the symmetric and exterior algebras. In section 5 he states analogous results for types \(B_ n\), \(C_ n\), \(D_ n\).
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 15A72
zbMATH DE Number: 3922911
zbMATH Keywords: classical Lie algebras
zbMATH Keywords: stable multiplicity
zbMATH Keywords: tensor algebra
zbMATH Keywords: irreducible representations
zbMATH Keywords: isotypic component
zbMATH Keywords: symmetric algebra
zbMATH Keywords: exterior algebras
Wikidata QID: Q115368994
author: Philip J. Hanlon
reviewed by: Yurii V. Mukhin
MaRDI profile type: Publication
full work available at URL: 0001-8708(85)90035-0
OpenAlex ID: W2037733454
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description: scientific article; zbMATH DE number 3922911